As mudasobwa's comment says, your using the case statement backwards. And wrongly.
You're checking if input contains letters that are not n or N, and if that's true, you delete stuff. So I could type foobar, and it'd delete files. Bad.
Even if I type exit or quit or whatever else to stop the task, it'll happily delete stuff. Even typing cancel would be interpreted as "yes, please delete stuff", since - though it contains an "n" - it contains lots of letters that are not "n".
You also have a (dead) branch for not typing y or Y, though that makes no sense. And you don't have an else branch - not that you need one, but it'd make more sense.
You have a fixme you haven't fixed. Your indentation is missing or wrong. Your message ("About to remove the tmp/letter_opener folder.") doesn't match what the code does.
In all, it's not great.
All you need is this:
namespace :cleanup do
desc 'Deletes the emails inside tmp/letter_opener folder.'
task letter_opener_emails: :environment do
puts "Remove tmp/letter_opener? [yn]"
if STDIN.gets.chomp =~ /^y$/i
FileUtils.rm_r("tmp/letter_opener")
else
puts "Aborting"
end
end
end